Fast Multiplierless Approximations of the DCT with the Lifting Scheme Jie Liang, Student Member, IEEE, and Trac D
3032 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 12, DECEMBER 2001 Fast Multiplierless Approximations of the DCT With the Lifting Scheme Jie Liang, Student Member, IEEE, and Trac D. Tran, Member, IEEE Abstract—In this paper, we present the design, implementation, the sparse factorizations of the DCT matrix [12]–[17], and and application of several families of fast multiplierless approx- many of them are recursive [12], [14], [16], [17]. Besides imations of the discrete cosine transform (DCT) with the lifting one-dimensional (1-D) algorithms, two-dimensional (2-D) scheme called the binDCT. These binDCT families are derived from Chen’s and Loeffler’s plane rotation-based factorizations of DCT algorithms have also been investigated extensively [6], the DCT matrix, respectively, and the design approach can also [18]–[21], generally leading to less computational complexity be applied to a DCT of arbitrary size. Two design approaches are than the row-column application of the 1-D methods. However, presented. In the first method, an optimization program is defined, the implementation of the direct 2-D DCT requires much more and the multiplierless transform is obtained by approximating effort than that of the separable 2-D DCT. its solution with dyadic values. In the second method, a general lifting-based scaled DCT structure is obtained, and the analytical The theoretical lower bound on the number of multiplications values of all lifting parameters are derived, enabling dyadic required for the 1-D eight-point DCT has been proven to be 11 approximations with different accuracies. Therefore, the binDCT [22], [23]. In this sense, the method proposed by Loeffler et al.
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